Question: Solve the equation. $\dfrac{dy}{dx}=-\dfrac{e^x}{8}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=-\dfrac{e^{-8x}}{8}+C$ (Choice B) B $y=-\dfrac{e^x}{8}+C$ (Choice C) C $y=-\dfrac{Ce^{-8x}}{8}$ (Choice D) D $y=-\dfrac{Ce^{x}}{8}$
Answer: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=-\dfrac{e^x}{8} \\\\ -8\,dy&=e^x\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} -8\,dy&=e^x\,dx \\\\ \int -8\,dy&=\int e^x\,dx \\\\ -8y&=e^x+C_1 \\\\ y&=\dfrac{e^x+C_1}{-8} \\\\ y&=-\dfrac{e^x}{8}+C \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=-\dfrac{e^x}{8}+C$